Subgroups — #18037

Order 256.

327 conjugacy classes of subgroups.

The subgroup lattice has 249 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2971	C₂	1	1	yes	maximal, characteristic, complemented
2	#2970	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#135	C₂	1	1	yes	maximal, characteristic
4	#1590	C₂	1	2	yes	maximal, complemented
5	#1584	C₂	1	2	yes	maximal, complemented
6	#1581	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆	C₂	1	2	yes	maximal, complemented
8	#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	#1488	C₂	1	1	yes	maximal, characteristic, complemented
10	#1488	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	C₄	1	1	yes	characteristic
13	C₄×#41	C₂²	1	1	yes	characteristic
14	C₄×#41	C₂²	1	1	yes	characteristic
15	#446	C₂²	1	2	yes	—
16	C₈×Q₈	C₂²	1	2	yes	—
17	C₄×Q₁₆	C₄	1	1	yes	characteristic
18	C₄×Q₁₆	C₂²	1	1	yes	characteristic
19	C₄×SD₁₆	C₄	1	1	yes	characteristic
20	C₄×SD₁₆	C₄	1	1	yes	characteristic
21	C₄×SD₁₆	C₂²	1	1	yes	characteristic
22	C₄×SD₁₆	C₂²	1	1	yes	characteristic
23	C₄×D₈	C₄	1	1	yes	characteristic
24	C₄×D₈	C₂²	1	1	yes	characteristic
25	#436	—	2	2	no	—
26	#436	C₂²	1	2	yes	—
27	#435	—	2	2	no	—
28	C₈×D₄	C₂²	1	2	yes	—
29	#432	—	2	1	no	—
30	#432	—	2	1	no	—
31	#432	C₂²	1	1	yes	characteristic
32	#432	C₂²	1	1	yes	characteristic
33	#431	C₂²	1	1	yes	characteristic
34	#427	C₄	1	1	yes	characteristic
35	#416	C₄	1	1	yes	characteristic
36	#416	C₄	1	1	yes	characteristic
37	C₄×M₁₆	—	2	1	no	—
38	C₄×M₁₆	C₂²	1	1	yes	characteristic
39	C₂×#97	—	2	1	no	—
40	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
41	#335	C₂²	1	2	yes	—
42	#334	C₂²	1	2	yes	—
43	Q₈⋊C₈	C₂²	1	2	yes	—
44	Q₈⋊C₈	C₂²	1	2	yes	—
45	D₄⋊C₈	C₂²	1	2	yes	—
46	D₄⋊C₈	C₂²	1	2	yes	—
47	#322	C₂²	1	2	yes	—
48	C₈²	C₂²	1	2	yes	—
49	C₂×#41	C₂×C₄	1	1	yes	characteristic
50	C₂×#41	C₂×C₄	1	1	yes	characteristic
51	#135	—	2	4	no	—
52	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
53	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂×D₈	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆	—	2	2	no	—
57	C₂×M₁₆	D₄	1	2	yes	—
58	C₂²×C₈	—	2	2	no	—
59	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
62	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
63	C₄×Q₈	C₂³	1	1	yes	characteristic
64	C₄×Q₈	C₂³	1	1	yes	characteristic
65	C₄×D₄	—	2	1	no	—
66	C₄×D₄	—	2	1	no	—
67	C₄×D₄	—	2	1	no	—
68	C₄×D₄	—	2	1	no	—
69	C₄×D₄	C₂×C₄	1	1	yes	characteristic
70	C₄×D₄	C₂×C₄	1	1	yes	characteristic
71	C₄×D₄	C₂³	1	1	yes	characteristic
72	C₄×D₄	C₂³	1	1	yes	characteristic
73	#117	—	2	1	no	—
74	#117	—	2	1	no	—
75	#117	C₂×C₄	1	1	yes	characteristic
76	#117	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²	—	2	1	no	—
78	C₂×C₄²	—	2	1	no	—
79	C₂×C₄²	C₂³	1	1	yes	characteristic
80	#107	C₂×C₄	1	1	yes	characteristic
81	#107	C₂×C₄	1	1	yes	characteristic
82	#106	C₂×C₄	1	1	yes	characteristic
83	#106	C₂×C₄	1	1	yes	characteristic
84	C₄⋊C₈	—	2	2	no	—
85	C₄⋊C₈	—	2	2	no	—
86	C₄⋊C₈	—	2	1	no	—
87	C₄⋊C₈	C₂³	1	2	yes	—
88	C₄⋊C₈	C₂³	1	2	yes	—
89	#103	C₂×C₄	1	1	yes	characteristic
90	#103	C₂×C₄	1	1	yes	characteristic
91	#103	C₂×C₄	1	1	yes	characteristic
92	#103	C₂×C₄	1	1	yes	characteristic
93	#102	C₂×C₄	1	1	yes	characteristic
94	#102	C₂×C₄	1	1	yes	characteristic
95	#102	C₂×C₄	1	1	yes	characteristic
96	#102	C₂×C₄	1	1	yes	characteristic
97	C₂²⋊C₈	—	2	1	no	—
98	C₂²⋊C₈	—	2	1	no	—
99	C₂²⋊C₈	—	2	1	no	—
100	C₂²⋊C₈	—	2	2	no	—
101	C₂²⋊C₈	—	2	2	no	—
102	#97	—	2	2	no	—
103	#97	—	2	1	no	—
104	#97	—	2	1	no	—
105	#97	D₄	1	1	yes	characteristic
106	#97	D₄	1	1	yes	characteristic
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	C₂×C₄	1	1	yes	characteristic
109	C₄×C₈	C₂×C₄	1	1	yes	characteristic
110	C₄×C₈	C₂³	1	2	yes	—
111	C₄×C₈	C₂³	1	1	yes	characteristic
112	C₄×C₈	C₂³	1	1	yes	characteristic
113	C₄×C₈	C₂³	1	2	yes	—
114	#41	—	2	2	no	—
115	#41	—	2	2	no	—
116	#41	—	4	1	no	—
117	#41	—	4	1	no	—
118	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×D₄	—	2	1	no	—
121	C₂×D₄	—	2	1	no	—
122	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
123	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄	#41	1	1	yes	characteristic
133	Q₁₆	—	2	2	no	—
134	SD₁₆	—	2	2	no	—
135	SD₁₆	M₁₆	1	4	yes	complemented
136	D₈	—	2	2	no	—
137	M₁₆	—	2	4	no	—
138	M₁₆	—	4	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	—	4	4	no	—
145	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₈	C₂×D₄	1	2	yes	—
150	C₂×C₈	C₂×D₄	1	2	yes	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
156	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
157	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
158	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
159	C₂²⋊C₄	—	2	1	no	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₂²⋊C₄	—	2	1	no	—
162	C₂²⋊C₄	—	2	1	no	—
163	C₂²⋊C₄	—	2	1	no	—
164	C₂²⋊C₄	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	C₂²×C₄	1	1	yes	characteristic
172	C₄²	C₂×D₄	1	1	yes	characteristic
173	C₄²	#41	1	1	yes	characteristic
174	C₄²	C₂⁴	1	1	yes	characteristic
175	C₂³	—	2	1	no	—
176	C₂³	—	2	1	no	—
177	C₂³	C₄×D₄	1	1	yes	characteristic
178	Q₈	—	2	1	no	—
179	Q₈	—	2	1	no	—
180	Q₈	—	2	1	no	—
181	Q₈	C₂×M₁₆	1	2	yes	—
182	D₄	—	2	1	no	—
183	D₄	—	2	1	no	—
184	D₄	—	2	1	no	—
185	D₄	—	4	2	no	—
186	D₄	—	4	2	no	—
187	D₄	C₂×M₁₆	1	2	yes	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	C₄×D₄	1	1	yes	characteristic
205	C₂×C₄	C₄×D₄	1	1	yes	characteristic
206	C₂×C₄	C₄×D₄	1	1	yes	characteristic
207	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
208	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄	C₂×#41	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	2	2	no	—
214	C₈	—	4	2	no	—
215	C₈	—	4	2	no	—
216	C₈	—	4	2	no	—
217	C₈	C₂×M₁₆	1	2	yes	—
218	C₈	#135	1	4	yes	complemented
219	C₂²	—	2	1	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	1	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	1	no	—
239	C₄	C₂²×M₁₆	1	1	yes	characteristic
240	C₄	#568	1	1	yes	characteristic
241	C₄	C₂×#135	1	2	yes	—
242	C₂	—	4	1	no	—
243	C₂	—	4	2	no	—
244	C₂	—	8	1	no	—
245	C₂	D₄×M₁₆	1	1	yes	characteristic
246	C₂	C₄×#135	1	1	yes	characteristic
247	C₂	#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #2971	C₂	semidirect
2. #2970	C₂	semidirect
4. #1590	C₂	semidirect
5. #1584	C₂	semidirect
6. #1581	C₂	semidirect
7. C₈×SD₁₆	C₂	semidirect
8. #1568	C₂	semidirect
9. #1488	C₂	semidirect
10. #1488	C₂	semidirect
11. C₈×M₁₆	C₂	semidirect
135. SD₁₆	M₁₆	semidirect
218. C₈	#135	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#2971	2
#2970	2
C₄×#135	2
#1590	2
#1584	2
#1581	2
C₈×SD₁₆	2
#1568	2
#1488	2
C₈×M₁₆	2